In Section  we prove the upper and lower density estimates for the local decay of the energy \(\mathcal {F}\)  on any sequence of \(m\) -minimizers \((A_m,u_m) \in \mathcal {C}_m\)  (see Theorem REF ) by considering a local version of \(\mathcal {F}^\lambda \)  (see (REF )), adapting arguments of [1]}, [2]}, [3]}
to our setting with displacements paired with free-crystal regions, and paying extra care to the fact that \(\mathbb {C}\)  is possibly not constant (but in \(L^\infty (\Omega \cup S)\cap C^{0}(\Omega )\) ).
